Spring–damper equivalents of the fractional, poroelastic, and poroviscoelastic models for elastography

Abstract

In MR elastography, it is common to use an elastic model for the tissue's response in order to interpret the results properly. More complex models, such as viscoelastic, fractional viscoelastic, poroelastic, or poroviscoelastic ones, are also used. These models appear at first sight to be very different, but here it is shown that they may all be expressed in terms of elementary viscoelastic models.For a medium expressed with fractional models, many elementary spring–damper combinations are added, each of them weighted according to a long‐tailed distribution of time constants or relaxation frequencies. This may open up a more physical interpretation of fractional models.The shear‐wave component of the poroelastic model is shown to be modeled exactly by a three‐component Zener model. The extended poroviscoelastic model is found to be equivalent to what is called a non‐standard four‐parameter model. Accordingly, the large number of parameters in the porous models can be reduced to the same number as in their viscoelastic equivalents. While the individual displacements from the solid and fluid parts cannot be measured individually, the main use of the poro(visco)elastic models is therefore as a physics‐based method for determining parameters in a viscoelastic model.

Restrictions on wave equations for passive media

“…The Biot model predicts three wave modes: one shear mode, and two compressional modes. It has recently been shown that the shear wave solution is exactly equivalent to a Zener model [23]. However, the two compressional wave modes are only approximately equivalent to spring damper models.…”

“…A function f : (0, ∞) → R is a Bernstein function if f (t) ≥ 0, all derivatives exist, and the sign of the derivatives alternate as in (23). A Bernstein function has the following representation…”

Harnessing brain waves: a review of brain magnetic resonance elastography for clinicians and scientists entering the field

“…The inversion times for these techniques are generally within a clinically feasible timeframe suggesting potential to make research applications more accessible to clinical workflows. These results also motivate future work exploring the use of more advanced forward models for generating training data that are more representative of brain tissue mechanical behavior, such as existing viscoelastic, 112 and poroelastic models. 113…”

General guidelines for the performance of viscoelastic property identification in elastography: A Monte‐Carlo analysis from a closed‐form solution

“…In parallel, the damping ratio, ξ was varied from 0% to 10%, a range indicative of biological tissue. 30 For the Monte-Carlo study, 2000 identifications for the storage and loss modulus indicated in Equation ( 2) were performed, with μ and ξ then defined by Equation (7) and Equation (8). These identifications were performed based on the minimization of F via the Matlab function fmincon (version 2021b, The MathWorks, Inc.; Natick MA, USA).…”